/sci/ - Science & Math

>>12740671
>proving an equivalence relation
It's by definition of the construction retard.

>>12740671
when you eat all your crayons, they'll be nom nom one

>>12740671
1/inf=0

>>12740680
1+1=1 when you add two separate units of clay together, you get one unit of clay.

>>12740671
the statement "[math]\lim_{n\to\infty}\left( \frac{2^n-1}{20^n} \right) > 0[/math] if [math]n > 0[/math]" doesn't make sense

>>12740671
Now take a guess what happens to lim (2^n-1)/20^n

>>12740739
Yes it does.
You have a positive number being devided by another positive number.

>>12740746
A positive number.
1>0.TTTT...>0.9999... so 0.9999...=1 is false.

Maybe you should start by trying to find this [math]\infty[/math]

>>12740765
the statement as a whole doesn't make sense

Nice going.
[math]0.9999\dots = \sum_{n=0}^\infty\frac{9}{10^n} \implies 0.9999\dots > 9[/math]

>>12740765
The last limit is equal to zero. The thing you're taking the limit for is positive, true. But the numbers approach zero. Taking a limit is not "lets replace every n by infinity", it's more like asking "which number does this sequence approach in the long run?"

>>12740720
this faggot again. 1 clay lump + 1 clay lump is not the same as 1 + 1. kys nigger

> broken mathematical notation
> another shizo attempt at derailing basic calculus
> sum starting at n=0

Into the trash it goes

>>12740820
Add two lumps of clay together and you get one unit of clay, 1+1=1.
>>12740790
0.9999...<0.TTTT...<1
So 0.9999... = 1 is false.

>>12740782
There you go, fixed.
0.9999...=1 is still false, no matter what you want to believe.

>>12740671
>By the way 1+1=1

you go this way, sir. >>12740821

x = 0.99999....
10x = 9.99999....
10x - x = 9.9999.... - 0.9999....
9x = 9
x = 1 . EOP

>>12740852
Just because you do not understand something does not mean it is false.

>>12740988
see >>12740739

>>12740994
Nope, 0.9999... < 0.TTTT... < 1.
Your statement is false.

>>12740996
It does.
Divide a positive number bigger than zero by another and you get a positive number.

>>12740998
show wich part.

>>12740988
I'm not sure you understand what infinite means

>>12741003
See >>12740988
>>12741004
That is very vague coming from a mathematician.
Show me where I am wrong.

>>12741002
it doesn't make sense semantically

>>12741011
x and y both > 0
then x/y > 0
Simple.

>>12741015
that's not where the problem is

>>12741018
So where is the problem?

>>12741015
i > 0
1 > 0
1/i = -i
-i < 0

>>12741027
Complex numbers, who have nothing to do with reality.
Here I am using only decimals. Do not be dishonest.

>>12741030
Okay.
sum of all natural numbers ( positive ones) / 10 < 0

>>12741041
Made a small typo there pal.
Sum of all natural numbers / 10 > 0

>>12741020
the limit inequality doesn't have "n" as a free variable, it's quantified inside of it in a "for all" formula. the letter n in the sentence "if n > 0" is a different variable, it has no relation to the n in the limit whatsoever.
it's like writing a for-loop in an "i" variable and then using the same letter outside of the cycle. same name, but totally different variables.

>>12741051
lim n->infinity 2^n -1 = a positive number.
lim n->infinity 20^n = a positive number.
divide a positive number by another and you get a positive number > 0

>>12741054
>lim n->infinity 2^n -1 = a positive number.
wrong
>lim n->infinity 20^n = a positive number.
wrong
>divide a positive number by another and you get a positive number > 0
true and unrelated

>>12741063
Both are true, if you do not get the fundamentals of mathematics, there is not much I can do to help you.

>>12741070
no, lim n->infinity 2^n -1 is not a number and neither is lim n->infinity 20^n

>>12740980
What is that number, 0.TTTT in fraction form?

>>12741289
20/20

>>12741292
Exactly, but I want to see OP et al. justify this claim.
>0.TTTTT must be a rational number, by construction
>Therefore it may be expressed as a fraction
>What is that fraction?
I'd like to see OP address this.

>>12741299
>>0.TTTTT must be a rational number, by construction
>by construction
why do you think so?

>>12741336
If you're going to play finitist games, then a sum of finite rational numbers is rational, and always tractable. So somewhere in your algebra, you should be able to tell me precisely which rational number 0.TTTTTTT... equals.

>>12741347
the sum is not finite though

>>12741350
Then why are you treating the sum as finite?

>>12741357
I'm not op. but I want to know why you think it's a rational number by construction when it's explicitly defined as an infinite sum.

>>12741364
OP is treating the infinite sum as a finite sum. The point is that rules that apply to finite sums don't apply to infinite sums. For example Eulers sum of inverse squares. An infinite sum of rational numbers yields an irrational number.

OPs last limit, claiming it's positive, is using rules of finite algebra. If that's the game he's playing, then the "infinite" sum, which he's treating as a finite sum, must be rational and he must be able to tell me what that rational number is.

>>12740720
If you take one lump of clay and place it next to another, which is what adding is, you get two lumps, 1 + 1 = 2.
When you take one lump of clay and squish it into another one, which is what multiplication is, you get one lump of clay, 1*1=1.
You're trying to pass multiplication off as addition.

>>12740988
You don't understand how limits work or bad a shitposting.

[math]
\begin{align}
\sum_{n=1}^{\infty} \frac{9}{10^n} &= \lim\limits_{n \rightarrow \infty} \frac{10^n - 1}{10^n} \\
&= \lim\limits_{n \rightarrow \infty} \frac{1 - \frac{1}{10^n}}{1} \\
&= 1
\end{align}
[/math]

Also

[math]
\lim\limits_{n \rightarrow \infty} \frac{2^n -1}{20^n}
= \lim\limits_{n \rightarrow \infty} \frac{1 - \frac{1}{2^n}}{18^n}
= 0
[/math]

>>12740988
You don't understand how limits work or bad a shitposting.

[math]
\begin{align}
\sum_{n=1}^{\infty} \frac{9}{10^n} &= \lim\limits_{n \rightarrow \infty} \frac{10^n - 1}{10^n} \\
&= \lim\limits_{n \rightarrow \infty} \frac{1 - \frac{1}{10^n}}{1} \\
&= 1
\end{align}
[/math]

Also

[math]
\lim\limits_{n \rightarrow \infty} \frac{2^n -1}{20^n}
= \lim\limits_{n \rightarrow \infty} \frac{1 - \frac{1}{2^n}}{20^{n-2}}
= 0
[/math]

>>12740671
It's obvious that there is always a positive epsilon that belongs to the reals such that 0.9999... + epsilon = 1. I don't understand why would people claim otherwise.

>>12741557
wrong. firstly, your equation determines the value of epsilon uniquely, the word "always" makes no sense here. secondly, the value is zero.

>>12741557
I think the key here is that the limit of the series does converge to one. But the original statement does not include the limit. Apples and oranges.

>>12741554
damn. fucked up the last denominator but whatever, same logic.

>>12741564
>limit of the series does converge to one
a limit doesn't converge to anything, it's a constant number. learn analysis.

>>12740720
Yeah but it's a larger unit. If you compare that lump of clay to one of the original ones, it will have twice the volume, so they are not the same.

>>12741030
>Complex number
>nothing to do with reality
kek

>>12741564
Anon just used the geometric series to write the limit which makes no sense since the series already converges since x<1.

>>12741564
0.999... already is the limit you brainlet

>>12741597
I guess that's why then. Some people would think 0.999... would be for some n>>0 but finite. While others would see it as the limit when n goes to inf. It's a problem of notation and not explaining what ... means in this context.

>>12741608
No it's a problem of idiots not understanding what's explained to them. The ellipses indicates a recurring sequence that never terminates. If people interpret that as something else, that's a them problem and not a notation problem.

>>12741636
It's not standard notation. You would have to define it.

>>12741640
????? give me a single rigorous treatment of the decimal expansion which would define [math]0.a_1 a_2 a_3 \dots[/math] differently than [math]\sum_{n=1}^{\infty}\frac{a_n}{10^n} = \lim_{n\to\infty}\sum_{k=1}^n\frac{a_k}{10^k}[/math]

>>12741653
As I said, I could have defined it as the truncated series for some large undetermined n. It's perfectly valid as long as I define it first.

>>12741656
>I could have defined it as the truncated series for some large undetermined n
now THAT is a non-standard notation

>>12741015
but if x_n/y-n > 0, it does not follow that lim x_n/y_n > 0. it's >= 0, and in this case it is =0, as it is explained in every elementary calculus book, not sure if you saw one of those.

>>12741608
nope, it's standard notation, no need to explain it. you could ask us to explain what the dot between the first two digits means.
whoever isn't educated enough to know conventions like this has no business posting in these threads.

>>12741656
>perfectly valid
How so? The decimal definition of a rational number is either a terminating sequence or a repeating sequence. Every definition I've ever seen has used this. I've never heard of your proposal for decimal number notation.

>>12741640
it is absoulte,y standard notation, alongside with 0.(9)

>>12741681
>whoever isn't educated enough to know conventions like this has no business posting in these threads.
this

>>12741681
You do understand that this recurring problem only stems from the lack of context provided right? Once you agree on the definition, there is no margin of doubt what the result is. Hopefully this will be the last time we see threads like these.

>>12741656
>I could have defined it as the truncated series for some large undetermined n
you might want to consult the dictionary term for 'rigorous'.

>>12741684
You'd be surprised that different scientific communities have different conventions for things. Perhaps you only read papers in your niche area and are not aware of this...

>>12741686
there is standard definition and there is nonstandard, and when a standard definition exists, a nonstandard one has to be given explicitly; in its absence we assume the standard definition. also, non-standard definitions have to be rigorous and consistent.

>>12741698
my niche area is mathematics. what other "niche area" has a definition for this particular notation?

>>12741705
Well then take your gay mathematics elsewhere

>>12741705
Even within mathematics there can be slighly different conventions, depending on the branch you focus. Even more so, when you start including physics, computer science, all branches of engineering and so on... All of this is science. There is a reason this board is called /sci/ and not /math/.

>>12741656
How would you define and differentiate rational and irrational numbers?

>>12741715
All the conventions are formally equivalent though.
0.(9) = 0.9... = [math]0.\bar{9} [/math]
They are different representations of the same definition. What is being proposed is a different definition of decimal numbers, which I don't see the utility for.

>>12741718
>being pedantic
No because that is not what is causing trouble. Have you never asked yourself why threads like this are being posted almost daily? It's precisely because there was no consensus on what ... was within this community. But like most academics, you'd rather argue about non practical things than addressing the real problem at hand...

>>12741698
>>12741686
>there's no standard definition of a decimal expansion
stop being retarded pls

You guys really like arguing about the same things over and over again.

>>12741725
>Have you never asked yourself why threads like this are being posted almost daily? It's precisely because there was no consensus on what ... was within this community.
please tell me you're joking

>>12741725
You're criticizing a mathematician for being pedantic? Ugh, compliment taken, I guess. It's a valid question for me to ask. You're proposing a non-standard decimal definition. I'm completely justified in asking you to explain basic properties in this proposed system. Or does rigor not matter to you?

>>12741731
Once you agree that is the limit, there is no other solution other than it being equal to 1. Threads like these wouldn't continue to be posted. What is so hard to understand?

>>12741734
I think you are just mad that you never realized what the real problem was. You were too autistic to see the point of view of someone else.

>>12741740
Okay so I'm autistic. Can you explain how your proposed decimal definition differentiates rational numbers from irrational numbers? Can you at the very least formalize it? After all, you're attempting to use it to construct a proof. You should formalize all the rules so we're all on the same page. For example, if you take a finite sum of rational numbers, then the sum is a unique rational number, which is well-defined. So what is this rational number that you propose exists between 0.99.. and 1?

>>12741715
we aren't talking general philosophy. we are talking about the meaning of 0.999... and since we have this thread every day for decades now, someone would already have brought up a legitimate alternate definition if it existed in any field.

>>12741725
>Have you never asked yourself why threads like this are being posted almost daily?
because people are trolling. but in this case it is really easy to make it improfitable because the topic is so clear, that trolls are quickly reduced to shit like this >>12741712

>>12740671
Your a stupid fuck

>>12741740
I'm trying to help you dude. I've always been bothered by the non-uniqueness of decimal representation of numbers. If alternate system exists that forces every decimal representation to be unique, I'd be elated. If you can formalize such a system and prove to me it's valid, I'm on your team. But if you can't even formalize any rigor into your statement, why should I believe it?

>>12741745
I can define a mathematical symbol as I like it. It overrides the standard notation. For instance, if I say * is now the convolution and no longer multiplication, then that is what it is. There is no real need to go full autismo. It's that simple. In this case, I could simply define ... as a FINITE decimal expansion, as I said.

>>12741740
we know the real problem is brainlets are envious of smart people and are hell-bent upon proving themselves to possess what they think is some sort of superiority e.g. look ma, they reply to my retardation.

>>12741755
I think it started off as a troll, but them idiots took the bait and actually believed it. Never underestimate the depths of human stupidity.

>>12741761
>I could simply define ... as a FINITE decimal expansion, as I said.
Okay so 0.999... is a finite sum of rational numbers. What's that rational number that it sums to, and can you prove its the smallest such number between 0.999... and 1?

>>12741761
then >>12741700 applies. either state your 'definition' or use the conventional one.
but in this case you cannot do even that. try it. define notation like 0.999... in a rigorous way. make the effort - I certainly won't make the effort to show your errors because the fact that no such definition was forthcoming in the history of these threads makes it infinitesimally improbable that such a definition exists. these threads aren't about an alternative definition anyway.

>>12741725
start your own thread where you carefully explain in the OP that the dots stand for the sum of an infinite series which itself is the limit of the sequence of partial sums. let's see how many 1 > 0.999... believers you convince.

>>12741761
>I could simply define ... as a FINITE decimal expansion
not rigorous and not what is the subject of the OP.

>>12741776
there are no such believers. there are those who know what the standard meaning is and there are those who pretend not to.

>>12741773
If you replace 0.999... by that truncated series (call it S), then there would be an epsilon > 0 such that 0.999... + epsilon = 1. Thus, proving that 0.999... < 1. epsilon = 1 - S > 0. All of this seems elementary. Again, this would only work if you define it as a finite decimal expansion.

>>12741786
with this definition 0.999... is not a well-defined real number

>>12741778
The title is 0.999... < 1. And I'm saying that is it possible to define ... as a finite decimal expansion such that statement is true.

>>12741786
>this would only work if you define it as a finite decimal expansion.
this is obviously not what OP talks about or he would not have carpetbombed us with badly formatted limits left and right.

>>12741790
The OP never claimed it was. This is why it's important to give enough context.

>>12741791
yes, for example we can define 0.999... = 0. you're not saying anything of value.

>>12741796
Again, context is all that matters.

>>12741795
do you also mention that "*" stands for multiplication and not for convolution every time you use it?

>>12741791
definitions don't look like that. is it a number? because it is not unique. is it the set of all such numbers? then it cannot be compared to a n7mber. etc. etc. etc., I understand you just heard about this little thing called arbitrariness of definitions but definitions have to be coherent and there is a huge body of conventional definitions which you can only bypass by explicitly stating your alternative definitions and proving that they are worth considering. this flailing about 'some finite decimal expansion' does not cut it, see my questions at the beginning of the post.

>>12741800
When there margin of doubt, I have to.

>>12741803
No one ever claimed that it had to be unique. Yes, it's a set of rational numbers. But that's beside the point.

>>12741795
>The OP never claimed it was
OP literally defined 0.999... as a limit which equals 1

>>12741812
Tbf the OP is a brainlet. I'm here just trying to put an end to these stupid baits. Or perhaps they can come up with better ones after reading this discussing. Lack of originality plagues all of 4chan. It feels like arguing with bots sometimes...

>>12741810
whoever claims it denotes a number e.g. by comparing it to a number does in fact claim it is unique.

>>12741831
You could compare sets...

>>12741835
Or you could have said, for any element in this set, the following holds. There is always a way...

>>12741835
there is no standard notation using '<' to compare sets, so it also would have to be stated in OP. and again, you could claim that all those letters claim something totally else and the whole OP is actually about Bar-Hillel's first theorem, but it would not be true.

>>12740671
Hey, OP, how is that limit greater than 0?

>>12741849
you fucked up the second line big time, though.

>>12741862
Yeah, but if OP can make up math, then so can I. My bullshit looks neater than his, so I'm right.

>>12741865
hmmm, the approach sounds promising...
(that's not an infinity of 'g's at the end.)

why do mathoid ppl do this

>>12740988
The second to last line is gibberish. The limit is equal to 0.

>>12740671
why, just fucking why do bait threads always get 100+ replies

>>12741849
2^n - 1 > 0 no matter what n > 0 you take.
Stop being dishonest.

>>12741081
lim n->infinity 2^n -1 > 0
and lim n->infinity 20^n > 0

>>12741444
Separating those lumps of clay seems great, but the purpose of adding them is to ensure that they are no longer separated. So instead of having 1 1 we have 1+1=1.

>>12742002
Just to be clear, what you're saying is lim n->infty 1/n > 0?

>>12742002
that doesn't mean they're positive numbers

>>12741996
What are you talking about? I merely define the > operator to mean possibly equal to 0. It's not my fault you guys don't understand context. Using > to mean equal to 0 is a standard notation.

>>12742018
Of course 1/n > 0, only 0/(whatever_you_want)=0.
Man these days scientists and mathematicians sure lack discipline.
>>12742019
Of course this does, what happened to common sense?

>>12741996
how is that relevant?

>>12742032
∞ is not a number

>>12742039
+∞ is positive.
-∞ is negative.
Simple.

>>12742040
it's positive, but it's not a number

>>12742040
+0 is positive
-0 is negative
Is 0 positive or negative?

>>12742046
0 is neither positive nor negative.
A positive number is greater than 0.
A negative number is inferior to 0.

>>12742062
Exactly. Simply because I can track a directional limit via a + or - symbol, it doesn't mean +0 or -0 are distinct numbers, or even numbers at all.

>>12742032
>Of course 1/n > 0
That is not what he asked.

Is lim n->infty 1/n > 0?

I have yet to see one single person show me that 0.9999... < 0.TTTT... < 1 is false.

>>12742075
Yes, only 0/(whatever_you_want)=0.

>>12742080
Because you're ignoring everybody who says you're wrong and simply state the same false argument over and over.
We told you where you're wrong.

>>12742072
Well, when we devide a positive number(2^n -1)
by another(20^n) we will get a positive number.
Simple.

>>12742086
can you define [math]\lim_{n\to\infty}\frac{1}{n}[/math] for us?

>>12742111
[math] \lim_{n \to \infty} \frac{1}{n} [/math]

>>12742111
0

>>12742157
>>12742148
>>12742111
A positive number close to zero but superior to zero.

>>12742381
is this number strictly lesser than 1/n for all n?

>>12742092
Poor mathematicians are losing touch with reality.
lim n->infinity 2^n - 1 > 0
lim n->infinity 20^n > 0
Divide a positive number by another and you get a positive number.
Soon they will be claiming that adding two lumps of clay do not result in a lump of clay just because 1+1=1 cannot be true in their minds.

>>12741983
Because the truth is natural to us all.
People are waking up and breaking free from blind obedience and faith in authority.
Even mathematicians are usually trained and domesticared people, not what true scientists should be.

>>12740671
There's no thorough notation of real number and all are approximations because our poor apeish brains only understand integers. 1,000000..... and 0,999999999... are right and left approximations.

>>12741557
People lack common sense. Especially scientists.

>>12742482
i saw this comic spammed a lot recently and finally decided to read through and it holy shit whoever wrote/posts that is absolutely delusional
it feels like a strawman of the "ugh im special everyone else is an npc" position that pervades 4chan, but seems completely sincere. embarassing

>>12743076
Same, but my interpretation was slightly different. It starts off fairly well grounded. Most people are followers, whilst some are leaders. Then it starts drifting into schizo territory.

>>12743081
the extremely anglo-centric view that most people are fine with the system and just follow it while making their own lives is obviously true in the west, where most 4chan users live. what rubbed me the wrong way was the essentialist, absolutist and still completely unfocused views portrayed here. literally just "some people are NPC normie poopoopeepees and thats just the way it is, and us based (propably racist) chads must rise up and fight muh system"
the system is broken but by alienating your fellow lemming you sure as hell arent gonna solve that

>>12742086
>Yes
Wrong.

>only 0/(whatever_you_want)=0.
Limit n-> inf of 1/x = 0

>>12742394
>Divide a positive number by another and you get a positive number.
The limit isn't a division, so this is irrelevant.

>>12742394
>lim n->infinity 2^n - 1 > 0
>lim n->infinity 20^n > 0
Both are false, the limit doesn't exist.

>>12743224
the limits are ∞

>>12743276
>the limits are ∞
No, the limits DNE.

So can someone prove 0.9999... = 1 without using false logic like saying 10 x 0.9999... - 0.9999... = 9?

>>12743640
>false logic like saying 10 x 0.9999... - 0.9999... = 9?
What's false about that?

>>12743640
Do you have a problem with this proof, and if so, where is your problem?

>>12743663

>>12740994
>x = 0.99999....
>10x = 9.99999....
>10x - x = 9.9999.... - 0.9999....
>9x = 9
>x = 1

Shouldn't it be
>10x - x = 9.9999.... - 0.9999....
>9.1111....x = 9

>>12743723
1/9 = 0.1111... is obviously false. converting an infinite sum into a fraction is a wonderful failure.

>>12743815
>Implying that 0.99999... + 0.1111... =1
99 + 11?
Use wolfram alpha if you can't calculate it.

>>12743663
b = 0.9999...
10 x b = 9.999... is a result with one less digit than b since multiplying by 10 shifts the result by one digit.
As a result, 10 x b - 9 = 0.999... < b.
Hence, 10b - b = 9b = 8.999... since the 9.999... from 10b has one less digit than b=0.999..., so the difference between the two results in something below 9.

>>12742111
7.5

>>12742086
0/0 = 1
Basic calculous

>>12743921
Irrelevant in this case.
Stop being dishonest.

Reminder that most of those postgraduate scientists who lurk on this board fail to even understand that 1+1=1 when you add two lumps of clay together.

>>12740671
Anybody unironically participating in this discussion is actually to dumb to see why they or the other person are wrong

>>12743928
Are they homomorphic?

>>12743640
For whatever a < [1 - 0.99999999..(put as many 9 s=as you want)| you can come up with b < a and b < |1-0.999999 (some more 9 as previously)| . The distance between 1,0000... and 0.99999.... is "infinitely small". Real numbers can only be written as limits left hand or right hand, 1,000000... is right hand, 0.9999999.. is left hand.
That's due to the fact any interval of real numbers as small as you want nevertheless contains an infinity of real numbers and can be cut in an infinity of intervals who themselves contain infinite numbers, etc.

0.9999... > 1

>>12744000
Checked and kekd

>>12743890
>10 is holy
lol
in base 2, 0.1 is an 'infinite sum'
retard

>>12743947
>are they homomorphic
Only if they follow the hairy ball theorem.

>>12742148
[math] \displaystyle
\lim_{n \to \infty} \dfrac{1}{n}
[/math]
Optimized.

>>12743890
Why is it a failure?

[math] 1.000... = 1[/math]

Surely you don't have any problems with that?

>>12740671
I can do this proof more efficiently (with about 45% less waste) in base 11.

[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0. \bar{3} = 0. \bar{9}
[/math]

>>12743916
>10 x b = 9.999... is a result with one less digit than b since multiplying by 10 shifts the result by one digit.
False, they have the same amount of digits.

Its not clear to me that you even understand what a limit is

The more I read this the more it seems like a shitpost, but I'm disappointed in the anons trying to refute him by bringing external arguments. I mean, you're right, but if you want to convince someone you have engage with what THEY'VE said, not bring additional things to the table. Ie: Show them why they're wrong, not why you're right.

>>12740671
>>12740739
A key sign of mathematical fuckery is when you immediately stop calculating in order to give a verbal argument. Especially you can continue to calculate. You left off at:

[math]lim\;n \rightarrow infinity\;( \frac{2^n-1}{20^n} )[/math]

but we can continue to reduce that expression:

[math]lim\;n \rightarrow infinity\;( \frac{2^n}{20^n}- \frac{1}{20^n} )[/math]
[math]lim\;n \rightarrow infinity\;( \frac{1}{10} ^n - \frac{1}{20} ^n )[/math]
0 - 0

So the difference is 0.
0.999... = 0.TTT... = 1

This is perhaps even trivially obvious if we think about expressing "0.TTT..." in base twenty.

Let [math]a_1 = \frac{9}{10}[/math] and [math]a_{n+1} = \frac{1}{10}a_n + \frac{9}{10}[/math]. Then the sequence [math]\lim_{n\to\infty}a_n = .999\dots[/math] However, [math]a_n > \frac{n}{n+10}[/math] for all [math]n[/math] and given any [math]k < 1[/math], I can find [math]N\in\mathbb{N}[/math] such that [math]a_n > \frac{n}{n+10} > k[/math] for [math]n\geq N[/math].

1/2^n sum is one. Write it in base 2 and it's profit.

>>12740671
> lim n->infinity ((2^n-1)/20^n) > 0
you can't make this shit up

>>12743396
why would you deny yourself valuable information like that

>>12740671
'You can always find another number closer to one'
Top kek that retard hasn't realized that's like a basic feature of limits. It's never truly zero or infinity

>>12744786
what is never truly zero or infinity?

>>12744837
One

>>12743815
>10x - x = 9.111... *x
Anon, I-

>>12740980
That just means you are using irregular metrics where 1 lump = 1/2 unit, try it with SI units and see if you get the same results.

Having read the thread more, it's clear that this all stems from OP's refusal to concede that, 1/∞ = 0. Or rather, that the limit as the denominator approaches ∞ is zero.

I mean, isn't this axiomatic? If OP refuses to work under this assumption, then isn't he just proposing an alternative system of infinitesimal arithmetic?

>>12740671
>infinity without the absorptive property
Don't tell Tooker

>>12740671
0.9999999... > 1 because 0.9999999... is infinite.

The limit as n approaches infinity of (2^n -1)/20^n is 0. The second to last statement is false. Nice try OP, but the simplest part of the proof is wrong.

>>12744892
1+1=3
am I proposing alternative axiomatic system for arithmetics? no, I'm just being retarded

>>12740671
>By the way 1+1=1, when you add two lumps of clay together, you get one unit of clay.
You don't need to invoke dumb shit like clay for 1+1=1, you only need to define + as the boolean logical operator OR like computer gates and flip flops do.

>>12744942
Disingenuous at best.
If 1+1=3, then 1=2. 0=1, all finite numbers equal all other finite numbers, and we've essentially abandoned the peano axiom that says 1=/=0. This is still a logically consistent system, albeit a useless one.

What exactly do we lose from OP's system? How does it affect our interpretation of the real world? How does it affect our ability to solve practical problems?
Are "imaginary" numbers retarded? Based on what you've implied, I can see arguments both for and against.

If a system of infinites and infinitesimals is useless, how is asserting that there are no infinitesimals and only one infinite limit any LESS useless?

>>12742104
for all n of N+ 2^n-1 is a positive number.
The limit of it is infinity
lol, get filtered

>>12741761

>>12744973
1+1=3, correct
1=3-1, correct
3-1 = 2, not correct as it denies the axiom.

>>12745021
>not correct as it denies the axiom.
No it doesn't. It just necessitates that 2=1. Which is perfectly valid, because 1=0 (in conjunction with the other basic axioms of arithmetic) necessitates that all finite numbers equal one another. Like I said.

1/9 = 0.111...
+
8/9 = 0.888...
=
9/9 = 0.999...

>>12744973
>What exactly do we lose from OP's system?
what op's system?

>>12745183
Darn, if only OP made some kind of image describing a particular system so your question could be answered.

>>12745205
yeah, his picture is anything but a description of an axiomatic system

>>12745219
What do you think his defining of .999... and .TTT... are if not axiomatic?

>>12745235
They are, and both are equal to 1, unless OP redefines what limit of a sequence means. Which he doesn't.

>>12745099
Nah retard.
1 + 1 = 3 is true
1 = 3 - 1 is true
3 - 1 = 2 is true, only if the first statement is not. Meaning that first and third one are from two axiomatic systems and not one.

>>12745243
Axioms are self evident truths, you don't need to redefine something to use it as the basis of your logic.

>>12745260
His logic is wrong then, simple as

>>12745260
>Axioms are self evident truths
no you retard
Axioms are what they are, starting points from which a logic tree grows up.
If an axiom isn't 'true' in the real world, it is useless for engineers.
But mathematically it can be fine. That math then just doesn't describe our reality.

>>12745273
>But mathematically it can be fine.
It is still a self evident truth (by definition) if it is completely abstract rather than empirical, if it is not a self evident truth ie something taken to be true on its own merit, even in the abstract sense, it is not used as an axiom and is not even useful mathematically or logically.

>>12745279
>I have literally no clue how modern math and logic work, the post

>>12745266
Then how does .TTTT... actually compare to .9999... if his analysis is wrong?

>>12745282
They're both 1 and thus equal

>>12745284
Then what step of the proofs did his logic and yours diverge?

>>12745298
Fourth line from bottom, the limit is not >0, but =0. Moreover, he didn't give any reason for this, he just stated it.

>>12741592
Nope.

>>12744776
>limits are division and inf/inf > 1 because I said so
>valuable

>>12742482
>Because the truth is natural to us all.
This is true.

Academia tries to gate keep wonder and discovery.

Science is a faith as much as chakras and voodoo.

>>12744965
Dumb shit? You have to be dumb not to understand 1+1=1 in regards to clay.

>>12745492
"lim n->∞ 2^n - 1 = ∞" is saying much more than "lim n->∞ 2^n - 1 does not exist"

>>12745508
What's the point? Why think about sums like this its useless and not relevant to anything

what did he mean by this?

>>12745564
He meant Riemann's hypothesis is false, god is dead and there are finitely many primes.

[math]
\displaystyle \sum_{n=1}^{\infty}\frac{9}{10^n} = \frac{9}{10} + \frac{9}{100} +\frac{9}{1000} + ...
= b = \frac{1}{10}(9 + \frac{9}{10} + \frac{9}{100} +...)
= \frac{1}{10}(9+b) = b \\
\frac{1}{10}(9+b) = b \\
\frac{9}{10} = \frac{9}{10}b \\
1 = b
[/math]

>>12745584
>>12744148
9.9999... - 0.9999... = 9 requires doing an infinite number of substractions.
Mathematicians have become deeply irrational.
lim n->infinity 2^n - 1 > 0 is true since 2^n > 1 if n > 0.
Only one operation is required in my case.

>>12744579
Falsehood and lies.
If (1/10)^n > (1/20)^n if n > 0 then
(1/10)^n - (1/20)^n > 0.
Utter failure from scientists at coming up with anything except con logic.
Scientists are con men these days.

>>12745635
prove that 2^n > 1 implies lim n->infinity 2^n - 1 > 0

>>12745563
The point is that 1+1=2 is not always true.
The point is that the spiritual is what matters, not the literal.
The meaning is what matters, not the words used.
1+1 =1, when you add two lumps of clay together, you get one lump of clay.

>>12745642
Sure.
+infinity > 0
2^n > 1 if n > 0
lim n->+infinity 2^n - 1 > 0

>>12744892
1/infinity > 0.
Only 0/(whatever_you_want_other_than_zero) = 0.

>>12745635
>one operation is required in my case
lobotomy

>>12745648
1/inf=0

>>12744033
When you divide something by another, the only true result appears when there is no infinite division.
4/2 = 2. Simple.
On the other hand 1/9 requires an infinite amount of division.
1/9 = 0.1111... is obviously false since the division processus has not ended yet and will never end.
Imagine me saying 20000 / 5 = 4...
That is exactly how flawed your logic is.

>>12745647
that's not a proof, sweaty

>>12745659
>>12744020

>>12745682
>>12745687
You have yet to show me that 0.999.. < 0.TTT... < 1 is false.

>>12740980
>playing with semantics is fun :)
go to /x/ if you want to twist words and meanings around for shits and giggles

>>12745724
T is a letter

>>12745724
[math]\lim_{n\to\infty} \frac{10^n - 1}{10^n} = 1[/math]
[math]\lim_{n\to\infty} \frac{20^n - 1}{20^n} = 1[/math]

>>12745644
Thats just a way another way of looking at it, its not necessarily important at all let alone more important than any other way of looking at it. Both are correct if you're not an idiot and you're a hack

>>12744973
What is OPs system? He's failed to provide any rigorous statements surrounding it, so how can we do anything about it? I asked him a simple question multiple times that he says is just me being pedantic. OP claims it's a finite sum of rational numbers. This means the finite sum must be a unique rational number, so what is that number?

>>12745659
>Imagine me saying 20000 / 5 = 4...
You can do that. 4.0000... = 4. Do you object to this?

I never thought it was even possible to seriously think 1 = 0.999...
No surprise these days boy = girl = trans = bi now.
The age of equivalence is a sad one, and it has infected mathematicians as well.

>>12746538
amuse us with your proof that 0.999... is not 1

>>12743903
0.999... * 10 < 9.999...

(1*10) - (0.000...001*10) = 8.999...?

10x -0.999... > 9x

10x -1+ 0.000...001= 9.000...001 *x?

>>12745254
How? How does "3-1=1" contradict "3-1=2"? Where's the contradiction?
All it does is imply that "1=2", which, while stupid, is logically consistent. It implies a system where all finite numbers equal all other finite numbers.

>>12745183
>>12746102
Fair, OP didn't give a system, and seems like a brainlet. But what he's getting at seems to suggest one.
He asserts both that lim(∞){1/n} equals a number, AND that said number isn't 0.
However, such a number then must be infinitely small. Lets call it "α"

Moreover, if it's not 0, then it doesn't have 0's multiplication properties. Ie:
2 * 0 = 0 , BUT, 2 * α =/= α

But how do we define α, or 2α, or any such variation? OP also implies this by how he constructed "0.TTT"
Whereas "0.999..." is 9 * 0.111... = 9 / [math]\sum_{n>0} 10^n[/math] , "0.TTT" is 19 / [math]\sum_{n>0} 20^n[/math]
For these to be different numbers, as OP asserts, that implies that [math]\sum_{n>0} 10^n[/math] =/= [math]\sum_{n>0} 20^n[/math]
This means that we have two infinite values that do not equal each other.

We thus construct a system of infinitesimals "α"s, defined as the inverses of varying infinites ω, defined by different infinite sums.
This all lies on the assumption that infinite sums equal a definite number, and not all such infinite numbers are equal.

The standard way of writing limits subtlety treats "∞" as a singular quantity. Even though infinity isn't a number. Infinitude is a property of a quantity. And in this case, the property of a new kind of number, the ω's.

>>12746786
Did someone say something about assigning infinity different values?

>>12746786
Until you convert it back from schizo base and realize they are the same number.

>>12746786
Ugh, I fucked this up.
The [math]\sum_{n>0} 10^{n} [/math] terms should read [math]\sum_{n>0} 10^{-n} [/math]

Just consider what I wrote in >>12744579 and note the step where I contend that the infinite power of a fraction less than 1 equals 0.
Instead, consider that it may instead equal a unique infinitesimal.

>>12746835
The whole point is that you axiomatically assume that they're not the same number, and thus have to construct a whole new category of numbers.

In the sense that any number times 0 equals 0, any finite number divided by ∞ equals 0. This is what the standard way of interpreting limits does. Thus, all infinities must have equal magnitude. But if you work on the ideas that not all infinities have equal magnitude, then their inverses must also equal numbers of varying magnitude.

>>12740671
How the hell is this still going on?

>>12740684
1/inf > 0 so it can't be equal to 0.
Only 0/a, with a=/=0 equals 0

>>12747281
>1/inf > 0
wrong

>>12747307
0/(whatever_you_want_not_equal_to_zero) = 0

>>12749286
and?

>>12749286
x/inf=0

>>12749319
unless x=inf

>>12745543
Right, it's saying "I don't instance what a limit is."

Given [math]n \in \mathbb{N}[/math] for [math]\frac{9}{10^{n}} > \frac{19}{20^{n}}[/math]
[math]\lim_{n\to\infty}\frac{19}{20^{n}} - \frac{9}{10^{n}} = 0[/math]
They both approach 0 with the same slope, so [math]0 = \lim_{n\to\infty}\frac{19}{20^{n}} = \lim_{n\to\infty}\frac{9}{10^{n}} = \lim_{n\to\infty}\frac{x-1}{x^{n}} \forall x \in \mathbb{N}[/math]

>>12749431
no, it's saying that 2^n - 1 grows beyond all finite bounds, in other words it has a limit equal to ∞ in [-∞,∞]

>>12749448
ignore the first line, I'm high on pizza and even used > instead of <
the rest stands

>>12740671
>1+1=1, when you add two lumps of clay together, you get one unit of clay.
That's not addition. The binary operator you've described is the union of the indicator function for clay.

>>12740790
This

>>12740671
If you all aren't using the syntax of the appropiate first order mathematical theory, how do you expect to reach a conclusion about decimal numbers without using significant figures?

>>12740720
This is perfectly legitimate. You can define your adding-clay operation and it will work as advertised.

However, you're talking about an operation that's distinct from the usual addition on natural numbers.

>>12740720
>benis+benis=benis
Isn't this just 0+0=0?

>>12752275
more like inf+inf=inf

>>12740680
Based

>>12753162
no, it's nothing like that at all

>>12753256
∞ + ∞ = ∞
a lot of stuff + a lot of stuff = a lot of stuff

>>12753267
>a lot of stuff + a lot of stuff = a lot of stuff
Brainlet.
[math]\frac{1}{0}= \infty[/math]
[math]+ \infty = - \infty[/math]
[math]\infty + \infty = \infty - \infty[/math]
Therefore,
[math]2 \infty = 0[/math]
>additive and multiplicative absorption is a lie

>>12753328
I have no idea what's your point

>>12753348
>I have no idea what's your point
You disgust me.

>>12753354
no, that wasn't the point of your post. what was it?

>>12753328
>anon now realizes why infinity is not a real number
I'm impressed this thread is still going

>>12753381
>why infinity is not a real number
?
not at all. All it shows is that [math]n \cdot \infty = 0[/math] when n is even, and [math]n \cdot \infty = \infty[/math] when n is odd.
The follow up question, "what about when 'n' is non-integer?" leads directly to the hyperreals.

You are the only one giving up on numbers when their definitions are right in front of you.

>>12753403
this algebra doesn't describe addition of clay at all

>>12749449
>no, it's saying that 2^n - 1 grows beyond all finite bounds
Then the limit DNE.

>>12753328
1/inf = 0
1 + inf = inf
1 - inf = -inf
inf + inf = inf
inf/inf undefined
inf - inf undefined
1^inf undefined

you can't do everything with inf as with a number, doesn't mean you can do nothing tho

>>12753588
you can absolutely 100% use it in the hyperreals (where (inf+1) - (inf+2) = -1, definitively), and as I proved earlier it is also compatible with wheel theory.

There is no ambiguity, besides the modulo 2 I mentioned here >>12753328.
We no longer need to fear the undefined.

>>12753694
>hyperreals, wheel theory
we're talking about -inf, R, +inf
you know, like normal people do

>>12753567
it does in [-inf,inf]

>>12740671
you don't have to construct obtuse mathematical formulas if you want to argue the definition of ellipsis or infinity.

>>12753694
unrelated to clay

>>12740671
This is retarded, lim n->∞ (2^n-1)/(20^n) is literally 0. Don't use math language in a math proof if you don't even know what the math language means.

>>12740720
it's 11, learn to concatenate

>>12754493
Which is two in base 1.

>>12753928
No, it doesn't. The limit would be a number that you can make 2^n-1 arbitrarily close to. You cannot make it arbitrarily close to infinity even if you treat infinity as a number, since 2^n-1 is always infinitely far from infinity as n approaches infinity.

>>12754498
idiot

>>12754566
Two concatenated ones is the representation of two in base one, you frog brained piece of shit. Kill yourself.

>>12754691
heh you fell for my bait

>>12749319

Infinity isn't a number, it's a concept. 0/inf has no answer. It's undefined.

NOW, if you were to say lim n->inf (0/n) that would be zero because you are evaluating the each positive value up to infinitely many values, all of which would be equal to, and head towards, zero.

However, 0/concept doesn't make any sense.

>>12740671
I'm willing to sacrifice myself to help you find the answer
I am 17+0.(9) years old
Mods will decide my fate

>>12754718
Silly me, you were just pretending to be retarded so you could post a .jpg of yourself.

>>12754813
That is exactly what I meant to imply. Good job anon.

>>12754817
Np.

>>12754823
kys

>>12754562
>You cannot make it arbitrarily close to infinity even if you treat infinity as a number
yes, you can. a sequence gets infinitely close to its limit in the sense that the sequence is eventually in every neighborhood of the limit. neighborhoods of ∞ in [-∞,∞] are sets in the form (x,∞]. their intersection is precisely ∞. so a sequence has limit ∞ if and only if it's eventually in (K,inf] for every real K, which is if and only if the sequence grows eventually beyond every lower bound K. which is exactly the definition of lim a_n = ∞ you find in any analysis or calculus textbook.

https://en.wikipedia.org/wiki/Extended_real_number_line#Order_and_topological_properties

>>12754783
False, infinity is a number greater than any number mankind can imagine.
Taking away the number propriety of infinity takes away all meaning from it.
infinity > any positive number you imagine.
-infinity < any negative number you imagine.
0/infinity = 0. Infinity is still a number and can occasionally be used, like in this case.
Operations between infinities cannot be used in the case of opposites operation, like infinity - infinity and infinity / infinity,
but for things like 0xinfinity and 0 / infinity, it certainly can. Both answers yield 0.
lim z->infinity(1/z) = 0+, the + is here to show you that no matter what y > 0 you can think of, y > 0+ > 0.
lim z->infinity(-1/z) = 0-, the - is here to show you that no matter what y < 0 you can think of, y < 0- < 0.
Simple.

>>12754490
if infinity > 0,
lim n -> infinity 2^n - 1 > 0 since 2^n -1 > 0 if n > 0
lim n - > infinity 20^n > 0 too.
So 0.9999... < 0.TTTT... < 1 no matter what mathematic propaganda was indoctrinated unto you.

>>12755621
>lim z->infinity(1/z) = 0+, the + is here to show you that no matter what y > 0 you can think of, y > 0+ > 0.
This is your mind on schizophrenia.
Stop talking about imagining things and be prezise

>>12754783
numbers are also "a concept" you brainlet. polynomials are not numbers and yet you can divide polynomials.

>>12755587
>a sequence gets infinitely close to its limit in the sense that the sequence is eventually in every neighborhood of the limit.
No, that has nothing to do with limits. The limit is inf if and only if for every real number ε > 0, there exists a natural number N such that for all n > N, we have |2^n-1 − inf| < ε.

Since |2^n-1 − inf| = inf for all n, this fails to hold for any ε, let alone every ε. Thus inf is not the limit.

>>12755621
>lim n -> infinity 2^n - 1 > 0
Yes, you have this part right. It has nothing to do with what lim n->∞ of that denominated by 20^n means in math language.

Aside from that, you failed to establish your base-20 lettering system clearly. T could very well be a digit that stands for 19, but in that case you've elided 10 other random letters between 9 and T, and failed to indicate which ones. Your whole concept is shabbily dressed, and I hope you remedy that before you post the same .png again tomorrow.

>>12755733
please read something about topological spaces. limits are a more general concept than you think.

>>12755733
did you even look at the wikipedia article posted? "Using this characterization of extended-real neighborhoods, the specially defined limits for x tending to +∞ and -∞ and the specially defined concepts of limits equal to +∞ and -∞ reduce to the general topological definition of limits." you've just shown that the metric on R can't be extended to [-∞,∞], the topology can be though.

>>12755876
Please read the definition of a limit until you understand it.

>>12755898
>did you even look at the wikipedia article posted?
Yes. It has nothing to do with limits. Just because something in some other context is called the same name doesn't mean it's the same thing.

>>12755986
L is the limit of f : X -> Y at p if for every open neighborhood V of L there exists an open neighborhood U of p such that U is mapped into V, with the possible exception of p.

>>12756021
it's the same context you undergrad. substitute. open neighborhoods of real numbers are open intervals. neighborhoods of infinity are open positive half-rays. substitute into the above for U and V and you get your babby definition of limits from calculus.

all of this is standard

>>12755642
False, 0+ is used to illustrate that a limit is above 0 and that it is superior to 0.
The difference between 0+ and 0- is a very important one, since it can mean something is either positive or negative.

>>12756141
Are you the same guy as OP? You're also not using math language correctly. "Open neighborhood" has no common semantic value. Neither does "neighborhoods of infinity," or "open positive half-rays." None of your argot is standard, let alone all of it.

>>12756094
>L is the limit of f : X -> Y at p if for every open neighborhood
Wrong.

>it's the same context you undergrad
No.

>>12756383
>You're also not using math language correctly. "Open neighborhood" has no common semantic value.
https://mathworld.wolfram.com/OpenNeighborhood.html
https://proofwiki.org/wiki/Definition:Open_Neighborhood
https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)#Neighbourhood_of_a_point
Neither does "neighborhoods of infinity,"
https://proofwiki.org/wiki/Definition:Neighborhood_of_Infinity
or "open positive half-rays."
https://en.wikipedia.org/wiki/Order_topology

>>12756451
>Wrong
sure bro

>>12756383
Truth is what matters most, not "common semantic value".